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A Liouville theorem of degenerate elliptic equation and its application
An equivalent characterization of the summability condition for rational maps
| 1. | School of Mathematics and Information Science, Henan University, Kaifeng 475004, China |
References:
| [1] |
H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.
doi: 10.1007/s00222-007-0108-4. |
| [2] |
H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306. |
| [3] |
P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46.
doi: 10.1017/S0143385700001802. |
| [4] |
J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96.
doi: 10.1007/s002220050239. |
| [5] |
J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415.
doi: 10.1007/s00222-008-0152-8. |
| [6] |
O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296.
doi: 10.1112/plms/pdn055. |
| [7] |
H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176.
doi: 10.4064/fm198-2-6. |
| [8] |
H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677.
doi: 10.1007/s11425-010-3134-4. |
| [9] |
H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799.
doi: 10.1093/imrn/rns105. |
| [10] |
T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680.
doi: 10.1007/s002220050236. |
| [11] |
F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403. |
| [12] |
F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178.
doi: 10.1016/j.ansens.2006.11.002. |
| [13] |
F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63.
doi: 10.1007/s00222-002-0243-x. |
| [14] |
J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636.
doi: 10.1017/S0143385706000629. |
| [15] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, preprint, ().
|
| [16] |
J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions,, preprint, ().
|
show all references
References:
| [1] |
H. Bruin, J. Rivera-Letelier, W. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.
doi: 10.1007/s00222-007-0108-4. |
| [2] |
H. Bruin, W. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306. |
| [3] |
P. Collet and J. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46.
doi: 10.1017/S0143385700001802. |
| [4] |
J. Graczyk and S. Smirnov, Collet, Eckmann and Hölder, Invent. Math., 133 (1998), 69-96.
doi: 10.1007/s002220050239. |
| [5] |
J. Graczyk and S. Smirnov, Non-uniform hyperbolicity in complex dynamics, Invent. Math., 175 (2009), 335-415.
doi: 10.1007/s00222-008-0152-8. |
| [6] |
O. Kozlovski and S. van Strien, Local connectivity and quasi-conformal rigidity of non-renormalizable polynomials, Proc. Lond. Math. Soc. (3), 99 (2009), 275-296.
doi: 10.1112/plms/pdn055. |
| [7] |
H. Li and W. Shen, Dimensions of rational maps satisfying the backward contraction property, Fund. Math., 198 (2008), 165-176.
doi: 10.4064/fm198-2-6. |
| [8] |
H. Li and W. Shen, On non-uniform hyperbolicity assumptions in one-dimensional dynamics, Science China Math., 53 (2010), 1663-1677.
doi: 10.1007/s11425-010-3134-4. |
| [9] |
H. Li and W. Shen, Topological invariance of a strong summability condition in one-dimensional dynamics, Int. Math. Res. Not., 8 (2013), 1783-1799.
doi: 10.1093/imrn/rns105. |
| [10] |
T. Nowicki and D. Sands, Non-uniform hyperbolicity and universal bounds for S-unimodal maps, Invent. Math., 132 (1998), 633-680.
doi: 10.1007/s002220050236. |
| [11] |
F. Przytycki, Hölder implies Collet-Eckmann, Géométrie complexe et systèes dynamiques (Orsay, 1995), Astérisque, 261 (2000), 385-403. |
| [12] |
F. Przytycki and J. Rivera-Letelier, Statistical properties of topological Collet-Eckman maps, Ann. Sci. Ecole Sup. Norm., 40 (2007), 135-178.
doi: 10.1016/j.ansens.2006.11.002. |
| [13] |
F. Przytycki, J. Rivera-Letelier and S. Smirnov, Equivalence and topological invariance of conditions for non-uniform hyperbolicity in the iteration of rational maps, Invent. Math., 151 (2003), 29-63.
doi: 10.1007/s00222-002-0243-x. |
| [14] |
J. Rivera-Letelier, A connecting lemma for rational maps satisfying a no growth condition, Ergodic Theory Dynam. Systems, 27 (2007), 595-636.
doi: 10.1017/S0143385706000629. |
| [15] |
J. Rivera-Letelier, Asymptotic expansion of smooth interval maps,, preprint, ().
|
| [16] |
J. Rivera-Letelier and W. Shen, Statistical properties of one-dimensional maps under weak hyperbolicity assumptions,, preprint, ().
|
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